Abstract
In this paper we propose an anisotropic extension of the isotropic exponentiated Hencky energy, based on logarithmic strain invariants. Unlike other elastic formulations, the isotropic exponentiated Hencky elastic energy has been derived solely on differential geometric grounds, involving the geodesic distance of the deformation gradient \({{\varvec{F}}}\) to the group of rotations. We formally extend this approach towards anisotropy by defining additional anisotropic logarithmic strain invariants with the help of suitable structural tensors and consider our findings for selected case studies.
Similar content being viewed by others
Notes
Note that not every objective and isotropic energy function can be expressed in terms of the logarithmic strain measures alone, see Neff et al. [48], whereas every such energy can be expressed in terms of the logarithmic strain tensor \(\log \varvec{U}\).
In the contracted notation the tensorial indices are allocated to the matrix indexes as follows \(\{11,22,33,12,23,13\}\rightarrow \{1,2,3,4,5,6\}\).
References
Anand L (1979) On H. Henckys approximate strain energy function for moderate deformations. J Appl Mech 46:78–82
Anand L (1986) Moderate deformations in extension–torsion of incompressible isotropic elastic materials. J Mech Phys Solids 34:293–304
Baker M, Ericksen JL (1954) Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids. J Wash Acad Sci 44:33–35
Ball JM (1977) Convexity conditions and existence theorems in non-linear elasticity. Arch Ration Mech Anal 63:337–403
Ball JM (2002) Some open problems in elasticity. In: Newton P, Holmes P, Weinstein A (eds) Geometry, mechanics, and dynamics. Springer, New-York, pp 3–59
Balzani D, Neff P, Schröder J, Holzapfel GA (2006) A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int J Solids Struct 43:6052–6070
Balzani D, Böse D, Brands D, Erbel R, Klawonn A, Rheinbach O, Schröder J (2012) Parallel simulation of patient-specific atherosclerotic arteries for the enhancement of intravascular ultrasound diagnostics. Eng Comput 29:888–906
Boehler JP (1978) Lois de comportement anisotrope des milieux continus. J Méc 17:153–190
Boehler JP (1979) A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. Zeitschrift für angewandte Mathematik und Mechanik 59:157–167
Brands D, Klawonn A, Rheinbach O, Schröder J (2008) Modelling and convergence in arterial wall simulations using a parallel FETI solution strategy. Comput Methods Biomech Biomed Eng 11:569–583
Bruhns OT, Xiao H, Meyers A (2000) Hencky’s elasticity model with the logarithmic strain measure: a study on Poynting effect and stress response in torsion of tubes and rods. Arch Mech 52:489–509
Bruhns OT, Xiao H, Meyers A (2001) Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky’s logarithmic strain tensor. Proc R Soc Lond Ser A Math Phys Eng Sci 457:2207–2226
Ebbing V, Schröder J, Neff P (2009) Approximation of anisotropic elasticity tensors at the reference state with polyconvex energies. Arch Appl Mech 79:651–657
Ehret AE, Itskov M (2007) A polyconvex hyperelastic model for fiber-reinforced materials in application to soft tissues. J Mater Sci 42:8853–8863
Flory PJ (1961) Thermodynamic relations for high elastic materials. Trans Faraday Soc 57:829–838
Ghiba ID, Neff P, Martin R (2015) An ellipticity domain for the distortional Hencky-logarithmic strain energy. Proc R Soc Lond Ser A Math Phys Eng Sci 471:20150510
Ghiba ID, Neff P, Šilhavỳ M (2015) The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity. Int J Non linear Mech 71:48–51
Glowinski R, Le Tallec P (1984) Finite element analysis in nonlinear incompressible elasticity. In: Oden J, Carey G (eds) Finite elements, Vol V: special problems in solid mechanics. Prentice-Hall, Englewood Cliffs
Glowinski R, Le Tallec P (1988) Augmented Lagrangian methods for the solution of variational problems. Springer, Berlin
Glowinski R, Le Tallec P (1989) Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, volume 9 of SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia
Hartmann S, Neff P (2003) Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int J Solids Struct 40:2767–2791
Hencky H (1928) Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Zeitschrift für technische Physik 9:215–220
Hencky H (1929) Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern? Zeitschrift für Physik 55:145–155
Hestenes M (1969) Multiplier and gradient methods. J Optim Theory Appl 4:303–320
Hill R (1968) On constitutive inequalities for simple materials. J Mech Phys Solids 16:229–242
Hill R (1970) Constitutive inequalities for isotropic elastic solids under finite strain. Proc R Soc Lond Ser A Math Phys Eng Sci 314:457–472
Hill R (1978) Aspects of invariance in solid mechanics. Adv Appl Mech 18:1–75
Hoger A (1987) The stress conjugate to logarithmic strain. Int J Solids Struct 23:1645–1656
Holzapfel GA (2006) Determination of material models for arterial walls from uniaxial extension tests and histological structure. J Theor Biol 238:290–302
Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48
Hyun SK, Nakajima H (2003) Anisotropic compressive properties of porous copper produced by unidirectional solidification. Mater Sci Eng A340:258–264
Itskov M, Ehret AE, Mavrilas D (2006) A polyconvex anisotropic strain-energy function for soft collagenous tissues. Biomech Model Mechanobiol 5:17–26
Jog CS (2006) Derivatives of the stretch, rotation and exponential tensors in n-dimensional vector spaces. J Elast 82:175–192
Jog CS, Patil KD (2013) Conditions for the onset of elastic and material instabilities in hyperelastic materials. Arch Appl Mech 83(5):661–684
Latorre M, Montáns FJ (2015) Anisotropic finite strain viscoelasticity based on the Sidoroff multiplicative decomposition and logarithmic strains. Comput Mech 56:503–531
Löblein J, Schröder J, Gruttmann F (2003) Application of generalized measures to an orthotropic finite elasto-plasticity model. Comput Mater Sci 28:696–703
Martin RJ, Neff P (2016) Minimal geodesics on \(\text{ GL }(n)\) for left-invariant, right-\(\text{ O }(n)\)-invariant Riemannian metrics. J Geom Mech 8(3):323–357
Mihai LA, Neff P (2017) Hyperelastic bodies under homogeneous Cauchy stress induced by three-dimensional non-homogeneous deformations (To appear in Mathematics and Mechanics of Solids)
Mihai LA, Neff P (2017) Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations. Int J Non Linear Mech 89:93–100
Montella G, Govindjee S, Neff P (2016) The exponentiated Hencky strain energy in modelling tire derived material for moderately large deformations. J Eng Mater Technol 138:031008
Neff P (2000) Mathematische Analyse multiplikativer Viskoplastizität. Ph.D. thesis. Technische Universität Darmstadt. Shaker Verlag, Aachen
Neff P, Ghiba ID (2016) The exponentiated Hencky-logarithmic strain energy. Part III: coupling with idealized isotropic finite strain plasticity. Contin Mech Thermodyn 28:477–487
Neff P, Mihai LA (2016) Injectivity of the Cauchy-stress tensor along rank-one connected lines under strict rank-one convexity condition. (To appear in Journal of Elasticity)
Neff P, Eidel B, Osterbrink F, Martin RJ (2013) The Hencky strain energy \(||\text{ log }~{U}||^2\) measures the geodesic distance of the deformation gradient to SO(3) in the canonical left-invariant Riemannian metric on GL(3). Proc Appl Math Mech 13:369–370
Neff P, Eidel B, Martin RJ (2014) The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky. arXiv:1402.4027
Neff P, Ghiba I, Lankeit J (2015) The exponentiated Hencky-logarithmic strain energy. Part I: constitutive issues and rank-one convexity. J Elast 121:143–234
Neff P, Lankeit J, Ghiba ID, Martin RJ, Steigmann DJ (2015) The exponentiated Hencky-logarithmic strain energy. Part II: coercivity, planar polyconvexity and existence of minimizers. Zeitschrift für angewandte Mathematik und Physik 66:1671–1693
Neff P, Eidel B, Martin RJ (2016) Geometry of logarithmic strain measures in solid mechanics. Arch Ration Mech Anal 222:507–572
Ogden RW (1972) Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc Lond A 326:565–584
Ogden RW (1997) Non-linear elastic deformations. Courier Corporation, Chelmsford
Powell M (1969) A method for nonlinear constraints in minimization problems. In: Fletcher R (ed) Optimization. Academic Press, New York, pp 283–298
Richter H (1948) Das isotrope Elastizitätsgesetz. Zeitschrift für angewandte Mathematik und Mechanik 28:205–209
Schröder J, Brinkhues S (2014) A novel scheme for the computation of residual stresses in arterial walls. Arch Appl Mech 84:881–898
Schröder J, Gross D (2004) Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Arch Appl Mech 73:533–552
Schröder J, Neff P (2003) Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int J Solids Struct 40:401–445
Schröder J, Gruttmann F, Löblein J (2002) A simple orthotropic finite elastoplasticity model based on generalized stress–strain measures. Comput Mech 30:48–64
Schröder J, Neff P, Balzani D (2005) A variational approach for materially stable anisotropic hyperelasticity. Int J Solids Struct 42(15):4352–4371
Schröder J, Neff P, Ebbing V (2008) Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. J Mech Phys Solids 56:3486–3506
Simo JC (1998) Numerical analysis and simulation of plasticity, vol. 6 of handbook of numerical analysis. Elsevier Science, New York
Spencer AJM (1987) Kinematic constraints, constitutive equations and failure rules for anisotropic materials. In: Boehler JP (ed) Applications of tensor functions in solid mechanics, volume 292 of CISM courses and lectures. Springer, Berlin, pp 187–197
Vallée C (1978) Lois de comportement élastique isotropes en grandes déformations. Int J Eng Sci 16:451–457
Acknowledgements
The first two authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 under the project “Novel finite elements for anisotropic media at finite strain” (SCHR 570/23-1), Sects. 1–4.3. Further, the first two authors would like to acknowledge support by the Deutsche Forschungsgemeinschaft within the framework of the project “Domain-decomposition-based fluid structure interaction algorithms for highly nonlinear and anisotropic elastic arterial wall models in 3D“ (SCHR 570/15-2) under the D-A-CH agreement, Sects. 4.4–5.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
1.1 Notes on the Hencky tensor
The Hencky strain tensor is defined through
and
The symmetric right Cauchy–Green tensor \({{\varvec{C}}}\) in spectral decomposition is given by
and for the Hencky strain we obtain
The first derivative of \(\log {{\varvec{U}}}\) with respect to \({{\varvec{C}}}\) can be computed as
Considering that
see for instance Jog [33], we find that
The second derivative for the linearization is given by
where
The exponent and the logarithm of an arbitrary symmetric tensor may also be be expressed with help of a Taylor expansion of the form
where the latter is convergent in a neighborhood of \(\mathbf{1}\).
1.2 Conjugate stress tensors
The following considerations are adapted from Ogden [50].
The constitutive equation for the stresses are derived form the (isothermal) entropy inequality
From the latter we deduce the constitutive relation \({{\varvec{P}}}= \partial _{{{\varvec{F}}}}\psi \). Let the generalized Lagrangean strain measures
and Eulerian strain measures
be given, we aim to find the corresponding constitutive equations. The so called stress power may be written as
where \({{\varvec{D}}}= \frac{1}{2}({{\varvec{L}}}+{{\varvec{L}}}^T)\) and \({{\varvec{L}}}= {\text {grad}}\dot{{{\varvec{x}}}}\). Considering that \(\dot{{{\varvec{E}}}}=\frac{1}{2}\left( \dot{{{\varvec{F}}}}^T{{\varvec{F}}}+{{\varvec{F}}}^T\dot{{{\varvec{F}}}}\right) \), we obtain the relations
The pairs in Eq. (88) are said to be work conjugate. By making use of the fact that \({{\varvec{R}}}^T\dot{{{\varvec{R}}}}= -\dot{{{\varvec{R}}}}^T{{\varvec{R}}}\), we may rewrite \(\dot{{{\varvec{E}}}}= \frac{1}{2}\left( {{\varvec{U}}}\dot{{{\varvec{U}}}}+\dot{{{\varvec{U}}}}{{\varvec{U}}}\right) \) and we are able to reformulate
such that we directly obtain the Biot stress \({{\varvec{T}}}_\mathrm{Biot}=\partial _{{{\varvec{U}}}}\psi ^\#({{\varvec{U}}})\), work conjugate to \({{\varvec{U}}}\) from the entropy inequality. With \({{\varvec{U}}}= {{\varvec{R}}}^T{{\varvec{V}}}{{\varvec{R}}}\) we are able to relate \({{\varvec{E}}}^{(m)}\) and \({{\varvec{K}}}^{(m)}\) and the corresponding time derivatives as follows:
Regarding the generalized stress-power it follows
Only if \({{\varvec{E}}}^{(m)}\partial _{{{\varvec{E}}}^{(m)}}\psi = \partial _{{{\varvec{E}}}^{(m)}}\psi {{\varvec{E}}}^{(m)}\), i.e. \(\partial _{{{\varvec{E}}}^{(m)}}\psi \) is coaxial with \({{\varvec{E}}}^{(m)}\), it immediately follows that the constitutive law results in
and the stress power is expressed through \(\langle \dot{{{\varvec{K}}}}^{(m)}, {{\varvec{R}}}(\partial _{{{\varvec{E}}}^{(m)}}\psi ){{\varvec{R}}}^T\rangle \), which is identical to Eq. (88).
The case that \(\partial _{{{\varvec{E}}}^{(m)}}\psi \) is coaxial with \({{\varvec{E}}}^{(m)}\) implies that also \(\partial _{{{\varvec{E}}}^{(m)}}\psi \) and \({{\varvec{U}}}\) are coaxial. Under this assumption one may show that
and it follows that
for isotropic materials. Inserting the latter result in Eq. (93) we obtain the relation
Regarding the conjugate stress to \(\log {{\varvec{U}}}\) the reader is also referred to Hill [26, 27], Hoger [28].
Rights and permissions
About this article
Cite this article
Schröder, J., von Hoegen, M. & Neff, P. The exponentiated Hencky energy: anisotropic extension and case studies. Comput Mech 61, 657–685 (2018). https://doi.org/10.1007/s00466-017-1466-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-017-1466-4